Peaceman-Rachford splitting for a class of nonconvex optimization problems

We study the applicability of the Peaceman-Rachford (PR) splitting method for solving nonconvex optimization problems. When applied to minimizing the sum of a strongly convex Lipschitz differentiable function and a proper closed function, we show that if the strongly convex function has a large enough strong convexity modulus and the step-size parameter is chosen below a threshold that is computable, then any cluster point of the sequence generated, if exists, will give a stationary point of the optimization problem. We also give sufficient conditions guaranteeing boundedness of the sequence generated. We then discuss one way to split the objective so that the proposed method can be suitably applied to solving optimization problems with a coercive objective that is the sum of a (not necessarily strongly) convex Lipschitz differentiable function and a proper closed function; this setting covers a large class of nonconvex feasibility problems and constrained least squares problems. Finally, we illustrate the proposed algorithm numerically

Self-cancellation of ephemeral regions in the quiet Sun

With the observations from the Helioseismic and Magnetic Imager aboard the Solar Dynamics Observatory, we statistically investigate the ephemeral regions (ERs) in the quiet Sun. We find that there are two types of ERs: normal ERs (NERs) and self-cancelled ERs (SERs). Each NER emerges and grows with separation of its opposite polarity patches which will cancel or coalesce with other surrounding magnetic flux. Each SER also emerges and grows and its dipolar patches separate at first, but a part of magnetic flux of the SER will move together and cancel gradually, which is described with the term "self-cancellation" by us. We identify 2988 ERs among which there are 190 SERs, about 6.4% of the ERs. The mean value of self-cancellation fraction of SERs is 62.5%, and the total self-cancelled flux of SERs is 9.8% of the total ER flux. Our results also reveal that the higher the ER magnetic flux is, (i) the easier the performance of ER self-cancellation is, (ii) the smaller the self-cancellation fraction is, and (iii) the more the self-cancelled flux is. We think that the self-cancellation of SERs is caused by the submergence of magnetic loops connecting the dipolar patches, without magnetic energy release.Comment: 6 pages, 4 figures, accepted for publication in ApJ